So the core of the matter is to develop sound recognition for the particular sound of the existing door bell, while preferably be as immune as possible to any other kind of sound, I suppose.
There are several ways of tackling this, one would be to start with the recognition problem using a PC and develop this on PC. Make several recordings of the door bell and analyze them with Python or your prefered language, then see how you can adapt this to a MCU (so it would have to be relatively simple). Use simple DSP algorithms.
One other approach, even if possibly overkill (or not?) could be to use some MCU able to run Tensorflow, and generate a model for recognizing this particular sound. That seems doable, but if you're not familiar with Tensorflow, there'll be some learning curve. Benefit, it can be called "AI" and will teach you some of that, if you're not familiar yet.
Another option would be to consider one of those "voice recognition" ICs, which may work well enough for recognizing some other sounds, like this:
https://www.wtsoundic.com/voice-recognition-ic/Although, the simplest approach is probably to detect activity of the door bell by placing a device nearby (doesn't need to be connected to it) that would sense some EMI that is probably emitted by it when it rings.
You could start by experimenting to see if it's workable: place some coil nearby the door bell unit, connect it to a scope, make the door bell ring and see if you can detect something useful. If so, you can then design some conditioning amplifier and link that to a MCU, if you want flexibility, or it could just trigger some other device that produces a loud sound, or flashes of light, or something.
Hope there's an idea that may help.